On perturbation method for the first kind equations: regularization and application

Let A lee a bounded operator in a Banach space X with range R ( A ) in a Banach space Y. Consider the following linear operator equation

Ax = f, f € R (A).(1)

We assume that the domain R ( A ) can be nonclosed and Ker A = { 0 } . In many practical problems one needs to solve an approximate equation

Ax = f,(2)

instead of exact equation. Here A and f are approximations of exact operator A and right-hand side function f is correspondingly such as

У — All < 61, Ilf — fII < 62, 6 = max{61,52}.(3)

The problem of solving of ecpiation (2) is ill-posed and therefore unstable even with respect to small errors and it needs regularization in real world numerous applications. The basic-results in regularization theory and methods for solving of the inverse problems have been gained in scientific schools of A.N. Tikhonov, V.I. Ivanov and M.M. Laverentiev. Nowadays, this field of contemporary mathematics promotes the developments of many interdisciplinary fields in science and technologies [1-4, 19, 21, 24]. There have been proposed many efficient regularization methods for operator ecpiation (1). The most efficient regularization methods are Tikhonov’s method of stabilizer functional, the quasisolution method suggested by V.K. Ivanov, M.M. Laverentiev perturbation method, V.A. Morozov’s discrepancy principle and other methods. Variational approaches, spectral theory, perturbation theory and functional analysis methods play the principal role in the theory . V.P. Maslov (see to [8]) has established the equivalence of existence of solution to ill-posed problem and convergence of the regularization process. There is a constant interest for regularization methods to be applied in interdisciplinary research and applications related to signal and image processing, numerical differentiation and inverse problems.

In this article we consider the regularized processes construction by introduction of the following perturbed equation

Ax _a + B ( a ) x _a = f.                               (4)

In this paper we continue and upgrade the results [2, 12, 18]. It is to be noted that regularization method based on perturbed equation was first proposed by M.M. Laverentiev [4] in case of completely continuous self-adjoint and positive operator A and B ( a ) = a.

Following [17] we select the stabilizing operator (SO) B ( a ) to make solution x _a unique and provide computations stability. Let us call a G S C R ⁿ as vector parameter of regularization. Here S is an open set, with zero belonging to the boundary of this set (briefly, S-sectoral neighborhood of zero in R ⁿ), lim B ( a ) = 0 . Parameter a we adjust S ∋ α → 0

to the data error level 5. Similar approach was suggested in the monographs [12,17], but in this article the regularization parameter a can be a vector. Previously only the simple case has been considered with B ( a ) = B0 + aB 1, a G R⁺. Such SO has been employed in the development and justification of iterative methods for Fredholm points A 0, zeros and the elements of the generalized Jordan sets of operator functions [6,11] calculation, for the construction of approximate methods in the theory of branching of solutions of nonlinear operator equations with parameters [13,15-17], for construction of solutions of differential-operator equations with irreversible operator coefficient in the main part [17]. In present article we propose the novel theory for operator systems regularization.

The paper is organized as follows. In Sec. 1 we obtained the sufficient conditions when perturbed ecpiation (4) enables a regularization process. In Sec. 2 we suggested the choice of SO B ( a ) . An important role is played by a classic Banach - Steinhaus theorem. In Sec. 3 we consider the application of regularizing equation of the form (4) in the problem of stable differentiation.

1.    The Fundamental Theorem of Regularizationby the Perturbation Method

Apart from ecpiations (1), (2), (4) introduce the ecpiations

( Ax + B ( a )) x = f,                               (5)

( Ax + B ( a )) x = f.                               (6)

Errors of operator B(a) can be always included into operator A. Equation (6) is call a regularized ecpiation (RE) for problem (2). The following estimates are assumed to be fulfilled below

||(A + B(a))-1||< c(|a|),(7)

l|B(a)II < d(lai),(8)

where c(lai) is a continue>us function, a G S C Rn, 0 6 S, lim c(lai) = to, lim d(|a|) = |α|→0

0 . If x* is a solution to equation (1). then ( A + B ( a )) - ¹ f — x* = ( A + B ( a )) - ¹ B ( a ) x*. Therefore, we have

Lemma 1. Let x* be some solution to equation (1), x ( a ) satisfy equation (4)- Then, in order to x_a ^ x* for S Э a ^ 0 it is necessary and sufficient to have the following equality fulfilled

S ( a,x* ) = || ( A + B ( a )) - ¹ B ( a ) x*|| G 0 for S Э a ^ 0 .               (9)

In [5] there are sufficient conditions for ensuring estimates (7) - (8), and examples addressing case of vector parameter. Application of such estimates for solving nonlinear equations are also considered. Let us follow [12] and introduce the following definition.

Definition 1. Condition (9) is called a stabilization condition. Operator B ( a ) , is called a stabilization operator if it satisfies the condition (9). Solution x* is calied a B-normal solution of equation (1).

Remark 1. Obviously the limit of the sequence {x_a} is unique in a normed space and therefore equation (1) can have only one B -normal solution.

From estimates (7) - (8) it follows

Lemma 2. Let x_a a nd x _a be solutions of equations (4) and (5) correspondingly. If parameter a = a ( 5 ) G S is selected such as 5 ^ 0

|a ( 5 ) | ^ 0 an id 5c ( |a ( 5 ) | ) ^ 0 ,                                (10)

then lim ||x_a — x_a|| = 0 .

δ→ 0

Definition 2. Condition (10) is called a coordination condition of vector parameter a with error level 5.

The coordination conditions play a principal role in all regularization methods for ill-posed problems (see, e.g. [2,4,7,10,12,17,19,23]). The coordination condition is assumed to be fulfilled. Below we also assume that a depends on 5 but for the sake of brevity we omit this fact.

Lemma 3. Let estimates (7) - (8) be satisfied as well as coordination condition for the regularization parameter (10). Next, we select q G (0 , 1) and find 5 >  0 such as for 5 <  5 0 the following inequality

^{5c (} ^ 1)) < q                                   (И)

holds. Then A + B ( a ) is a continuously invertible operator and the following estimates are fuilfilled

|| ( A + B ( a )) ^-11| < IIA i BA-l,                   ₍₁₂₎

1 —q

||(A + B(a))-1 f||< ||(A + B(a))-1 f|| + .5^||(A + B(a))-1 f||.(13)

¹ - q

Proof. Based on estimate (3) for arbitrary f we have

||(A - A)(A + B(a))-1 f || < 5||(A + B(a))-1 f ||.(14)

Hence taking into account estimates (7), (8), (11), we have the following inequality

||(A - A)(A + B(a))-1 f ||< 5c(|a|) < q||f ||.(15)

Now, since q <  1 we have A + B ( a ) = ( I + ( A — A )( A + B ( a )) - 1 )( A + B ( a )) . Then existence of inverse operator ( A + B ( a )) - ¹ , as well as estimate (12) follows from known inverse operator Theorem. Next, we employ the following operator identity C- ¹ = D^{- 1} — D^{- 1} ( I + ( C — D ) D^- 1 ) ^- 1 ( C — D ) D^{- 1} where C = (71 + B ( a )) , D = A + B ( a ) and, using-on inequalities (14), (15) we get estimate (13).

□ Theorem 1. [Main Theorem] Let conditions of Lemma 3 be fulfilled, i.e parameter a is coordinated with noise level 5. Then RE (6) has a unique solution x_a. Moreover if in addition x* is a solution of exact equation (1), then the following estimate is fulfilled

||x a — x* || < S ( a, x* ) + 5—q ) (1 + ||x* || + S ( a, x* f) .              (16)

If also x* is a B-normal solution of equation (1) then {x_a} converges to x* at a rate determined by (16) as 5 ^ 0.

Proof. Existence and uniqueness of the sequence {x_a} as a solution of RE (6) for a G S ."1                                                                  *1

was proved in Lemma 3. Since ( A + B ( a ))( x _a

^w

^1

— x* ) = f — f — ( A — A ) x*

l~W

^i

— B ( a ) x*, then ₁

we get the desired estimate (16) ||x _a — x*|| < || ( A + B ( a )) ^{- 1} 1| ( ||f — f || + || ( A — A ) x*|| + || ( A + B ( a )) ^{- 1} B ( a ) x*|| ) < S ( a,x* ) + ^- 0 1 ^1 + ||x*|| + S ( a,x* )^ based on estimates (12). (13) and (8). Since x* is a, B -normal solution then lim S ( a,x* ) = 0 . And thanks to a — 0

parameter a coordinated with noise level 5 we 1 nave lim 5c ( |a| ) = 0 . Hence, due to (16) S—— 0

lim ||x 1 _a — x*|| =0 which completes the proof. S — 0 □

As footnote of the section it’s to be mentioned that for practical applications of this theorem one needs recommendations on the choice of SO B ( a ) anc1 a B- normal solution existence conditions. It’s also useful to know the necessary and sufficient conditions of existence of a B -normal solution x* to the exact equation (1). These issues we discuss below.

2.    Stabilizing Operator B(a) Selection, B-Normal Solutions Existence and Correctness Class of Problem (1)

If A is a, Fredliolin operator, {^ i } П is a laasis in N ( A ) , {fi i } П is a laasis in N* ( A ) , then (see n

Sec. 22 in [22]) one may assume B ( a ) = fPfinZ^Zn wh^ere {d i }, {z i } ^are selected such as i =1

{1 if г = k herewith the equation

0 ii г = k

n

Ax = f - ^ f^ i i z i i =1

is resolvable for arbitrary source function f. Let us now recall f, which is a ^-approximation nn of f. Then perturbed equation Ax + J2(x, yi^zi = f — fifiZ-fiZi lias a, miiqire solution x i=1                 i=1

such as ||x — x*|| ч 0 ford ч 0, where x* is unique solution of exact equation (17) for which {x*,Yi} = 0, i = 1 ,n. Thus, in the case of a, Fredholm operator A as a, stabilizing n operator one can take a finite-dimensional operator B = J2(-,7i)zi which does not depend 1

on parameter a. That is the regularization of iterative methods we employed in our papers [9,15-17] for the study of the second order nonlinear equations with parameters. Of course, with this choice of SO B it is required to have information about the kernel of operator A and its defect subspace. Therefore, it is of interest to give recommendations on the choice of SO B ( a ) without the use of such information. In solving more complex problem it is important to consider the first kind equations, when the range of operator A is not closed. It is to be noted that in papers [10,18] and in monograph [12,17] we constructed the SO for the first kind equations as B о + aB 1 where a G R ⁺ . Below we consider the generalization of such results when B = B ( a ) , a G S C R ⁿ. Our previous results presented in papers [10,12,18] follow from the following theorems 2 and 3 as special cases.

Theorem 2. Let || ( A + B ( a )) - ¹ 1| <  c ( l a i ) , || B ( a ) || <  d ( l a i ) for a G S C R ⁿ, c ( |a| ) , d ( |a| ) be continuous functions, lim c ( |a| ) = to , lim d ( |a| ) = 0 . Let | a |v 0                        | a |^ 0

lim c ( |a| ) d ( |a| ) <  to , N ( A ) = 0 , R ( A ) = Y. Then the 'unique, solution x* of equation |a|v 0

(Ij is a, B-normal solution and operator B ( a ) is its SO.

Proof. First, let B ( a ) x* G R( A ) fc>r a G S. Then there exists an element x 1 ( a ) such as Ax i( a ) = B ( a ) x*. Theri ( A + B ( a )) ^{- 1} B ( a ) x* = ( A + B ( a )) ^{- 1} ( Ax ₁( a ) + B ( a ) x 1 ( a ) — B ( a ) x ₁( a )) = x ₁( a ) — ( A + B ( a )) ^{- 1} B ( a ) x ₁( a ) . Since B (0) = 0 , N ( A ) = { 0 } then lim x ₁( a ) = 0 . It is to lie noted that bv condition || ( A + B ( a )) ^{- 1} B ( a ) || <  c ( | a | ) d ( | a | ) , S ^ a ^ 0

where c ( | a | ) d ( | a | ) is a, continuous function such as lim c ( | a | ) d ( | a | ) is fruite. the a -sequerice | a |v 0

{|| ( A + B ( a )) ^{- 1} B ( a ) x* ||} is infinitesimal when S Э a ч 0 . The sequence of operators { ( A + B ( a )) ^{- 1} B ( a ) } converges pointwise to the zero operator on the linear manifold L ₀ = { x | B ( a ) x G R ( A ) } . Thus, we have proved that the theorem is true when B ( a ) x* G R ( A ) . By condition sup c ( | a | ) d ( | a | ) <  to tlre a -sequerice {|| ( A + B ( a )) ^{- 1} B ( a ) ||} is bounded.

α ∈ S

Therefore, the sequence of linear operators { ( A + B ( a )) ^{- 1} B ( a ) } in space X converges pointwise to the zero operator on the linear manifold L ₀ = { x | B ( a ) x G R ( A ) } . But then, on the basis of the Banach - Steinhaus theorem we have a pointwise convergence of this operator sequence to the zero operator on the closure L ₀ , i.e. when B ( a ) x* G R ( A ) . Since R ( A ) = Y arid B ( a ) G L ( X ч Y ) , then B ( a ) x* G Y and theorem 2 is proved.